Regular Families of Curves

A family F of non-intersecting curves filling a metric space is called regular if, in a neighborhood of any point >̂, it is homeomorphic with a family of straight lines. We have given in another paper a necessary and sufficient condition, which we shall call (A') (to be described below), that a family F be regular. We shall prove in this note that the following condition is sufficient : (A) Given any point p, and a direction on the curve through p, there is an arc pq in this direction with the following property. For every e > 0 there is a ô>0 such that for any p', with p(p', p) <à, there is an arc p'q' of C(p) such that